>For those who don't know it, in brief: Put a vertical pole in flat ground at noon and measure the length of the shadow. The next day, drive an hour or two directly north or south and do the same measurement at noon again. With a bit of trigonometry, you can estimate the earth's circumference - or for our purposes, at least prove to yourself that the earth is not flat, as the shadows will be different lengths with the sun in nearly the same position both times.

Sure, but a flat earther will just argue that the sun is much closer than scientists claim, which explains the difference in shadow length. And again, what's a layman to do?

The best experiment to prove the earth is round that I've thought up involves the Coriolis Effect and a rifle fired at different points both north and south of the equator. Unfortunately that requires international travel with firearms, making it legally difficult to try.

Personally, I like multiple Foucault pendulums as this demonstrates rotation and curvature. I think the part that is missing in everyone's explanation is how to prove it is a sphere and not a hemisphere. Obviously, going to the "edge" would immediately shows this is false. But I think the only way to do this without moving too much is to record lunar eclipses and do the math to figure out that you would eventually see straight edges in an eclipse if the earth was not completely spherical.

Interesting idea! The sun would have to be about 6'000 km away I think. The full moon / new moon / solar eclipse proves that the moon is much closer than the sun. The moon would have to be dangerously close. Then, we know that the moon causes tides, and we can measure them as well on different cities. Being so close, the sun would have to cause tides as well, unless it is very light. It would have to take some time to calculate all that...

The only thing that is missing here is that the different lengths don't matter by themselves. The explanation could be that the rays of the sun are not parallel. Of course, this would put the sun closer than the moon but we are assuming we cannot use extraterrestrial knowledge. To complete the logic, you need to measure in multiple locations and note that the shadows do not change as expected if there was no curvature of the earth.

Totally true. More in general, it is a relatively simple mathematical fact that a sphere without a point is homeomorphic (even conformally!) to a plane. Therefore, it is impossible to prove that the Earth is spherical unless you either do measurements and run numbers (like you propose) or visit every single location of it (including all the seas and both poles; particularly the south pole, which for some reason flat earthers tend to assume it is missing). The only way to do the latter, as far as I can tell, is to be in a sufficiently polar orbit for enough time (or at least have a satellite in orbit), and few people are able to do that (directly, i.e., without trusting somebody else).

It is interesting that is not uncommon for flat earthers to be more "scientific" that many spherical earthers in recognizing that many spherical Earth arguments are not sufficient by themselves. Of course the spherical earthers happen to be right, while the flat earthers then proceed to invent completely implausible theories to support their beliefs, but they deserve some recognition for the first stage.